Network theory is a way of describing the world in terms of a model called a network
that allows us to capture the information about the relationship between things.
But lets first think about why we might be interested in this at all.
We often describe the world in terms of objects or things and their properties, we talk about
countries and their GDP, people and their age or the color of a car, this type of component
based analysis works well when the system we are interested in is relatively isolated.
But when we turn up the interactions and connectivity between elements within a system it is increasingly
the connections that come to shape the elements and define the system as a whole and thus
we need a model that captures this information about the relationships and allows us to reason
about it, this is where network theory comes in.
Network theory starts with a very simple view of the world as made up of nodes which are
things or objects, like people, cities, computers etc. and the relationships between these things,
called edges, such as friendships, trading partners, cables and so on.
This abstract representation of the world, can be used to model a wide variety of things,
thus we can have social networks, biological networks consisting of interacting creatures
within an ecosystem or logistic networks composed of interacting suppliers and consumers.
Network theory gives us a set of tools for analyzing the individual elements and relations
within these networks, the structure of the network and the properties that these different
networks structures give rise to.
The first set of question we might like to ask about a particular network relate to its
degree of connectivity, that is how connected an individual element or the wholes network
is, this will tell us many things about it such as how quickly a new event could spread
or propagate through the system.
The average degree of connectivity will give us a quick answer to this; this is calculated
by taking the total number of edges and dividing it by the total number of nodes within the
network
We also need to take into account how large the network is, that is to say how far is
it on average from one point to another.
This is called the average path length and we can calculate it by taking the average
of all the path length between all the nodes.
Because networks are all about connectivity we often ascribe value to individual nodes
based upon their degree of connectivity, there are various methods for calculating this but
a popular one called Eigenvector centrality, which measures both how many edges a node
has and how connected the nodes it joined to are also.
Popular web search engines use variants of this Eigenvector centrality measure to rank
webpages by calculating both the number of links into a webpage and the degree of connectivity
of the pages that link into them thus gaining an idea of the relative importance of the
website
Next we are interested in talking about the overall structure to the network this will
be largely determined by how the relationship between the nodes was formed.
If the relations between elements was generated randomly we would expect a relatively even
distribution of edges across the network, this type of structure or topology is called
a distributed network as the relative importance of any node is distributed across the entire
network.
A second type of network structure we can get is called decentralized or small world,
this is generated by having local clusters of connections, but also having some random
distant connections.
an example of this might be a group of friends, with some of the friends having distant relatives
in other parts of the world.
By using these local connection within the group and distant connections research has
shown that it is possible to connect two random people within a average of just six steps
and thus it is termed small world.
Lastly we have more centralized networks called scale free networks, this is where may nodes
have chosen to connect to the same node giving it a degree of connectivity that greatly exceeds
the average whilst leaving may with a very low level of connectivity.
Many real networks are through to be scale-free, including social, biological and technological
systems such as world-wide web, where very few sites like Wikipedia have a very large
amount of links into them, whilst the vast majority of websites have very few.
These various types of network structures give rise to different properties, a key question
we are interest in asking here is how robust or fragile is a particular type of network
as this will not only help us understand networks better, but will also be of great significance
in how we design and manage them.
For example, think about a country with many small to medium size cities supplying the
population with various public services, if we were to remove one of the cities it would
have a limited effect on the overall system, because the networks has a distributed structure
making it robust to fail of this kind.
In contrary if we take a county with one dominant capital city with the rest of the urban network
dependent upon it for core services, this centralized network may be more efficient
but it is also in what is called a more critical state as effecting this single primary node
would have a large systemic effect.
As we transit from an industrial to information societies, networks are emerging as a new
paradigm in how we structure our systems of organization both social and technological.
Network theory is a young and rapidly growing area that provide us with a set of tools for
designing and managing these new types of organization and more generally understanding
the world around us from a different perspective.